Inspired abilities trigger no matter how the creature becomes untapped: by the turn-based action at the beginning of the untap step or by a spell or ability.

2/1/2014

If an inspired ability triggers during your untap step, the ability will be put on the stack at the beginning of your upkeep. If the ability creates one or more token creatures, those creatures won’t be able to attack that turn (unless they gain haste).

2/1/2014

Inspired abilities don’t trigger when the creature enters the battlefield.

2/1/2014

If the inspired ability includes an optional cost, you decide whether to pay that cost as the ability resolves. You can do this even if the creature leaves the battlefield in response to the ability.